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March  2008, 7(2): 317-353. doi: 10.3934/cpaa.2008.7.317

Global attractors for a three-dimensional conserved phase-field system with memory

Gianluca Mola 1,

1. 

Dipartimento di Matematica "F.Brioschi", Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

Received  January 2007 Revised  May 2007 Published  December 2007

We consider a conserved phase-field system on a tridimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature $\vartheta$. These effects are represented through a convolution integral whose relaxation kernel $k$ is a summable and decreasing function. Therefore the system consists of a linear integrodifferential equation for $\vartheta$ which is coupled with a viscous Cahn-Hilliard type equation governing the order parameter $\chi$. The latter equation contains a nonmonotone nonlinearity $\phi$ and the viscosity effects are taken into account by the term $-\alpha \Delta\chi_t$, for some $\alpha \geq 0$. Thus, we formulate a Cauchy-Neumann problem depending on $\alpha $. Assuming suitable conditions on $k$, we prove that this problem generates a dissipative strongly continuous semigroup $S^\alpha (t)$ on an appropriate phase space accounting for the past histories of $\vartheta$ as well as for the conservation of the spatial means of the enthalpy $\vartheta+\chi$ and of the order parameter. We first show, for any $\alpha \geq 0$, the existence of the global attractor $\mathcal A_\alpha $. Also, in the viscous case ($\alpha > 0$), we prove the finiteness of the fractal dimension and the smoothness of $\mathcal A_\alpha $.
Keywords: global attractor., Conserved phase-field models, memory effects, absorbing sets.
Mathematics Subject Classification: 35B41, 37L30, 45J05, 80A2.
Citation: Gianluca Mola. Global attractors for a three-dimensional conserved phase-field system with memory. Communications on Pure and Applied Analysis, 2008, 7 (2) : 317-353. doi: 10.3934/cpaa.2008.7.317
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